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On P vs. NP, Geometric Complexity Theory, and the Riemann Hypothesis
, 2009
"... Geometric complexity theory (GCT) is an approach to the P vs. NP and related problems suggested in a series of articles we call GCTlocal [27], GCT18 [30][35], and GCTflip [28]. A high level overview of this research plan and the results obtained so far was presented in a series of three lectures i ..."
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Geometric complexity theory (GCT) is an approach to the P vs. NP and related problems suggested in a series of articles we call GCTlocal [27], GCT18 [30][35], and GCTflip [28]. A high level overview of this research plan and the results obtained so far was presented in a series of three lectures in the Institute of Advanced study, Princeton, Feb 911, 2009. This article contains the material covered in those lectures after some revision, and gives a mathematical overview of GCT. No background in algebraic geometry, representation theory or quantum groups is assumed. For those who are interested in a short mathematical overview, the first lecture (chapter) of this article gives this. The video lectures for this series are available at:
On P vs. NP, Geometric Complexity Theory, Explicit proofs, and the Complexity Barrier
, 2009
"... Geometric complexity theory (GCT) is an approach to the P vs. NP and related problems. This article gives its complexity theoretic overview without assuming any background in algebraic geometry or representation theory. ..."
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Geometric complexity theory (GCT) is an approach to the P vs. NP and related problems. This article gives its complexity theoretic overview without assuming any background in algebraic geometry or representation theory.
Reducibility and completeness
 Algorithms and Theory of Computation Handbook
, 1999
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Circuit Principles and Weak Pigeonhole Variants
, 2005
"... This paper considers the relational versions of the surjective and multifunction weak pigeonhole principles for PV , # 1 and # 2 formulas. We show that the relational surjective pigeonhole principle for # 2 formulas 2 implies a circuit blockrecognition principle which in turn implies the ..."
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This paper considers the relational versions of the surjective and multifunction weak pigeonhole principles for PV , # 1 and # 2 formulas. We show that the relational surjective pigeonhole principle for # 2 formulas 2 implies a circuit blockrecognition principle which in turn implies the surjective weak pigeonhole principle for # 1 formulas. We introduce a class of predicates corresponding to polylog length iterates of polynomialtime computable predicates and show that over R 2 , the multifunction pigeonhole principle for such predicates is equivalent to an "iterative" circuit blockrecognition principle. A consequence of this is that if R 3 proves this circuit iteration principle then RSA is vulnerable to quasipolynomial time attacks.
Renormalization group approach to the P versus NP question
, 2008
"... This paper argues that the ideas underlying the renormalization group technique used to characterize phase transitions in condensed matter systems could be useful for distinguishing computational complexity classes. The paper presents a renormalization group transformation that maps an arbitrary Boo ..."
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This paper argues that the ideas underlying the renormalization group technique used to characterize phase transitions in condensed matter systems could be useful for distinguishing computational complexity classes. The paper presents a renormalization group transformation that maps an arbitrary Boolean function of N Boolean variables to one of N −1 variables. When this transformation is applied repeatedly, the behavior of the resulting sequence of functions is different for a generic Boolean function than for Boolean functions that can be written as a polynomial of degree ξ with ξ ≪ N as well as for functions that depend on composite variables such as the arithmetic sum of the inputs. Being able to demonstrate that functions are nongeneric is of interest because it suggests an avenue for constructing an algorithm capable of demonstrating that a given Boolean function cannot be computed using resources that are bounded by a polynomial of N.